For a single boundary subregion \(A\) , the max flow-min cut theorem directly indicates that one can find a thread configuration that can lock the specified boundary subregion (and its complement simultaneously). In other words, there exist thread configurations that can lock a set of boundary regions \(I = \left\lbrace  {{A_1},{A_{\rm {c}}}} \right\rbrace \) , and there is typically an infinite number of choices. However, one can further ask, can we find a locking thread configuration that can lock an arbitrary specified set of subregions simultaneously? The question becomes very nontrivial. Broadly speaking, it depends not only on the relative space position relations between these specified subregions, but also on the properties we assign to the bit threads, in particular, the precise definition of the thread density bound. Recently, the authors in [1]} investigated this issue in great detail. They proposed and proved several theorems on the existence of locking thread configurations in various situations. Here, we merely point out one interesting locking theorem. See figure REF (b) (For more existence theorems and detailed technical proofs of the theorems, see the original paper [1]}):
